1. Mariano Andrenucci Professor, Department of Aerospace Engineering, University of Pisa, Italy Chairman and CEO, Alta S.p.A, Via A. Gherardesca 5, 56121 Ospedaletto, Pisa, Italy e-mail: m.andrenucci@alta-space.com Advanced Course ”Electric Propulsion Concepts and Systems” ESA/ESTEC, Noordwijk, September 15-19, 2008 Fundamentals of Plasma Acceleration

2. Slide 2.2 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Plasma Thrusters Unified approach based on acknowledgement that different types of electric thruster can all be described as Plasma Thrusters Plasma?

3. Slide 2.3 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Plasma Thrusters Unified approach based on acknowledgement that different types of electric thruster can all be described as Plasma Thrusters small Debye length

4. Slide 2.4 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci low neutral collisionality Plasma Thrusters Unified approach based on acknowledgement that different types of electric thruster can all be described as Plasma Thrusters

5. Slide 2.5 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Plasma Thrusters Unified approach based on acknowledgement that different types of electric thruster can all be described as Plasma Thrusters plasma parameter large

6. Slide 2.6 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Plasma Thrusters Unified approach based on acknowledgement that different types of electric thruster can all be described as Plasma Thrusters Main implication: quasi-neutrality assumption We shall call Plasma Thrusters all devices in which the working fluid remains quasi-neutral throughout all phases of the process Hall Thrusters, Self-field MPD Thrusters and Applied Field MPD Thrusters belong in this cathegory This definition leaves out ion thrusters, which inherently involve charge separation as a basic feature of the acceleration process

7. Slide 2.7 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Momentum Equation Under very general assumptions we can obtain the following Momentum Equation for the generic species To generate thrust we must transfer momentum to a working fluid. How can momentum be transfered to a plasma ? change of momentum electromotive force Lorentz emf pressure, viscosity interactions with particles of the same type collisions Interaction with particles of other types

8. Slide 2.8 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Two-fluid Model By considering only the electronic and ionic components of the plasma the following two-fluid model is obtained small m, negligible friction between the two fluids Isotropic pressure terms non-isotropic (viscous) terms

9. Slide 2.9 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci By neglecting the electron inertial term in the second equation and with the substitution we finally obtain Two-fluid Model

10. Slide 2.10 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Generalized Ohm’s Law With the further useful substitutions from the electron equation we obtain the Generalized Ohm’s Law Hall’s emf thermionic emfback emf electric field

11. Slide 2.11 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Electric Field By rearranging the generalized Ohm’s law we obtain the expression for the self-consistent electric field in the quasi-neutral plasma Thermionic emfHall’s emfOhmic termBack emf Resistive heating is exploited in arcjets This is exploited in different ways in MPD and HET thrusters Usually small No useful contribution in the velocity direction No useful contribution in the velocity direction

12. Slide 2.12 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci B General Vector Diagram Momentum increase of the ion fluid Electric field contribution Collisional contribution Going back to the two-fluid model, let us visualize the vector diagram of fields and currents (neglect the pressure gradient contributions)

13. Slide 2.13 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci B General Vector Diagram Momentum transfer to the ions Electric field contribution Collisional contribution Electric field effect on the electron fluid in the ion comoving frame Electric field effect due to electrons relative velocity Collisional momentum loss Going back to the two-fluid model, let us visualize the vector diagram of fields and currents (neglect the pressure gradient contributions)

14. Slide 2.14 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci General Vector Diagram By combining and posing we finally have Lorentz force Thus, the vector diagram of fields and currents for the two-fluid model (neglecting the pressure gradient contributions) can be visualized as shown here

15. Slide 2.15 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Energy Equations which can be rewritten as Collisional terms Joule heating The increase in the flow directed kinetic energy can be obtained by taking the dot product of the momentum equations for the two species by and respectively:

16. Slide 2.16 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Energy Equations Adding up the two equations above the collisional terms cancel out, and we are left with Once again we can explicitly highlight the role of the overall Lorentz force. With a few passages we would obtain but it should be remembered that the increase in the ion fluid kinetic energy is either drawn from the energy transferred by the electrons through collisions, or from direct action of the electric field on the ions

17. Slide 2.17 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Power transfer efficiency Thus we see that - neglecting again the pressure gradient terms - the useful energy transfered to the plasma can utimately be computed in terms of power delivered by the electric field minus power dissipated as Ohmic heating. We are thus prompted to define a power transfer efficiency as Hall parameter and remembering that we finally obtain being is the angle formed by the Lorentz force with the local flow direction

18. Slide 2.18 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example I : large Hall parameter

19. Slide 2.19 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example I : large Hall parameter

20. Slide 2.20 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example I : large Hall parameter

21. Slide 2.21 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example I : large Hall parameter

22. Slide 2.22 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example II : Hall parameter ~ 1

23. Slide 2.23 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example II : Hall parameter ~ 1

24. Slide 2.24 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example II : Hall parameter ~ 1

25. Slide 2.25 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Example II : Hall parameter ~ 1

26. Slide 2.26 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Hall-effect Thrusters

27. Slide 2.27 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Self-field MPD Thrusters

28. Slide 2.28 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Applied-field MPD Thrusters

29. Slide 2.29 Fundamentals of Plasma Acceleration ESA/ESTEC, Noordwijk, September 15-19, 2008 Advanced Course: Electric Propulsion Concepts and Systems M. Andrenucci Applied-field MPD Thrusters Including effect of self-field